Does this series converge or diverge? I know 1 diverges, but what about 0; my guts are telling me it converges.?

the nth partial sum S_n = 0*n = 0

S_n = 0
lim n→∞

So it converges to 0.

If you want to use x, then you need x variable in the summation. It is absent.So you will have no series and the argument is meaning less. if you want to use sum over x, then x should have values 1,2, 3 etc. This problem is meaningless.

It converges.
It is always 0 and stays at 0 .
The sum is 0 always also.

remember any geometric series with r between 0<=x < 1
Its sums converges
this is a geometric series with r = 0

It is bounded, so it converges.